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Proton-Seconds
Ian Beardsley!
(University of Oregon, Department of Physics, 2021)!
!
Genesis Project California 2021
⋅
⋅
of 2 31
Abstract
The idea that life could be part of a Universal Natural Process has been around for some time,
perhaps starting with the astronomer Fred Hoyle. In my effort to show artificial intelligence and
biological life elements are mathematical constructs I had formed a new constant that was
interesting in itself. This short paper treats just it and what follows from it, since it has life
connected to more universal values such as Planck’s constant, the universal constant of
gravitation, the speed of light, the fine structure constant, the radius of a proton, and the mass
of a proton. A program is written that shows it has incredible properties that predict all of the
elements. It is suggested that its value is in design so a definition of design is used from
Buckminster Fuller’s Synergetics. Ultimately, in the end the constant seems to be in step with
the phases of the moon and the earth and they way they were divided-up by the Ancients to
give us the calendar we have today. All of this in terms to be connected to the structure of the
basis of geometry as founded on fundamental principles.#
of 3 31
Theory For Inertia
In order to present the elements as mathematical structures we need to explain the matter from
which they are made as mathematical constructs. We need a theory for Inertia. I had found
(Beardsley Essays In Cosmic Archaeology. 2021) where I suggested the idea of proton
seconds, that is six protons-seconds, which is carbon the core element of biological life if we
can figure out a reason to divide out the seconds. I found!
Equation 1. !
Where h is Planck’s constant 6.62607E—34 Js, r_p is the radius of a proton 0.833E-15m, G is
the universal constant of gravitation 6.67408E-11 (Nm2)/(kg2), and c is the speed of light
299,792,459 m/s. And t_1 is t=1 second. is the Sommerfeld constant (or fine structure
constant) is 1/137. The mass of a proton is .!
The fine structure constant squared is the ratio of the potential energy of an electron in the first
circular orbit to the energy given by the mass of an electron in the Bohr model times the speed
of light squared: !
!
Matter is that which has inertia. This means it resists change in position with a force applied to
it. The more of it, the more it resists a force. We understand this from experience, but what is
matter that it has inertia?!
I would like to answer this by considering matter in one of its simplest manifestations, the
proton, a small sphere with a mass of 1.6726E-27 kg. This is a measure of its inertia.!
I would like to suggest that matter, often a collection of these protons, is the three dimensional
cross-section of a four dimensional hypersphere.!
The way to visualize this is to take space as a two-dimensional plane and the proton as a two
dimensional cross-section of a sphere, which would be a circle.!
In this analogy we are suggesting a proton is a three dimensional bubble embedded in a two
dimensional plane. As such there has to be a normal vector holding the higher dimensional
sphere in a lower dimensional space. Thus if we apply a force to to the cross-section of the
sphere in the plane there should be a force countering it proportional to the normal holding it in
a lower dimensional universe. This counter force would be experienced as inertia. It may even
induce in it an electric field, and we can see how it may do the same equal but opposite for the
electron. Refer to the illustration on the following page…#
1
t
1
α
2
m
p
h4π r
2
p
Gc
= 6protons
α
m
p
= 1.67262E − 27kg
α
2
=
U
e
m
e
c
2
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#
of 5 31
!
!
!
I made a program that looks for close to whole number solutions, I set it at decimal part equal
to 0.25. You cam choose how may values for t you want to try, and by what to increments
them. Here are the results for incrementing by 0.25 seconds then 0.05 seconds. Constant to all
of this is hydrogen and carbon. The smaller integer value of seconds gives carbon (6 protons at
1 second) and the largest value integer values of seconds give hydrogen (1 proton at six
seconds) and outside of that for the other integer values of protons you run are at t>0 and t<1.
Equation 1 really has some interesting properties. Here are two runs of the program( depart is
just me verifying that my boolean test was working right to sort out whole number solutions):!
By what value would you like to increment?: 0.25!
How many values would you like to calculate for t in equation 1 (no more than 100?): 100!
24.1199 protons 0.250000 seconds 0.119904 decpart !
12.0600 protons 0.500000 seconds 0.059952 decpart !
8.0400 protons 0.750000 seconds 0.039968 decpart !
6.0300 protons 1.000000 seconds 0.029976 decpart !
4.0200 protons 1.500000 seconds 0.019984 decpart !
3.0150 protons 2.000000 seconds 0.014988 decpart !
2.1927 protons 2.750000 seconds 0.192718 decpart !
2.0100 protons 3.000000 seconds 0.009992 decpart !
1.2060 protons 5.000000 seconds 0.205995 decpart !
1.1486 protons 5.250000 seconds 0.148567 decpart !
1.0964 protons 5.500000 seconds 0.096359 decpart !
1.0487 protons 5.750000 seconds 0.048691 decpart !
1.0050 protons 6.000000 seconds 0.004996 decpart !
0.2487 protons 24.250000 seconds 0.248659 decpart !
0.2461 protons 24.500000 seconds 0.246121 decpart !
0.2436 protons 24.750000 seconds 0.243635 decpart !
By what value would you like to increment?: 0.05!
How many values would you like to calculate for t in equation 1 (no more than 100?): 100!
40.1998 protons 0.150000 seconds 0.199837 decpart !
30.1499 protons 0.200000 seconds 0.149879 decpart !
24.1199 protons 0.250000 seconds 0.119904 decpart !
20.0999 protons 0.300000 seconds 0.099918 decpart !
17.2285 protons 0.350000 seconds 0.228500 decpart !
15.0749 protons 0.400000 seconds 0.074938 decpart !
1
α
2
m
p
h4π r
2
p
Gc
= (6protons)(1secon d )
1
α
2
m
p
h4π r
2
p
Gc
= (1proton)(6secon d s)
α
2
=
U
e
m
e
c
2
of 6 31
12.0599 protons 0.500000 seconds 0.059950 decpart !
10.0500 protons 0.600000 seconds 0.049958 decpart !
8.0400 protons 0.750000 seconds 0.039966 decpart !
7.0941 protons 0.850000 seconds 0.094088 decpart !
6.0300 protons 1.000000 seconds 0.029975 decpart !
5.2435 protons 1.150000 seconds 0.243457 decpart !
5.0250 protons 1.200000 seconds 0.024980 decpart !
4.1586 protons 1.450000 seconds 0.158605 decpart !
4.0200 protons 1.500000 seconds 0.019985 decpart !
3.1737 protons 1.899999 seconds 0.173673 decpart !
3.0923 protons 1.949999 seconds 0.092296 decpart !
3.0150 protons 1.999999 seconds 0.014989 decpart !
2.2333 protons 2.699999 seconds 0.233325 decpart !
2.1927 protons 2.749999 seconds 0.192719 decpart !
2.1536 protons 2.799999 seconds 0.153564 decpart !
2.1158 protons 2.849998 seconds 0.115782 decpart !
2.0793 protons 2.899998 seconds 0.079303 decpart !
2.0441 protons 2.949998 seconds 0.044061 decpart !
2.0100 protons 2.999998 seconds 0.009993 decpart !
1.2433 protons 4.850000 seconds 0.243294 decpart !
1.2306 protons 4.900001 seconds 0.230607 decpart !
1.2182 protons 4.950001 seconds 0.218177 decpart !
Here is the code for the program:!
#include <stdio.h>!
#include <math.h>!
int main(int argc, const char * argv[]) {!
!
int n;!
float value=0, increment,t=0, p=1.67262E-27, h=6.62607E-34,G=6.67408E-11,
c=299792459,protons[100],r=0.833E-15;!
!
do!
{!
printf("By what value would you like to increment?: ");!
scanf("%f", &increment);!
printf("How many values would you like to calculate for t in equation 1 (no more than 100?): ");!
scanf("%i", &n);!
}!
while (n>=101);!
{!
for (int i=0; i<n;i++)!
{!
protons[i]=((137*137)/(t*p))*sqrt(h*4*(3.14159)*(r*r)/(G*c));!
!
int intpart=(int)protons[i];!
float decpart=protons[i]-intpart;!
t=t+increment;!
if (decpart<0.25)!
{ printf("%.4f protons %f seconds %f decpart \n", protons[i], t-increment, decpart);!
}}}}!
of 7 31
While I understand that one second is a human invention and can’t be taken as significant, the
equations have meaning in that there are two equations each utilizing a second so they are
connected. But what really makes me wonder is how one can predict carbon, the other
hydrogen so accurately with the unit of a second. It was a conspiracy on the part of those who
formulated the duration of a second a long time ago to be what it is? The Ancient Egyptians,
The Babylonians, The Julian calendar—who, what, when?!
6 protons gives a little more than a second. This makes a shorter day. We have!
Equation 2. !
H=1.00784 g/mol, carbon = 6 protons!
h=6.62607E-34, r_p=0.833E-15, G=6.67408E-11, c=299,792,459!
(24hours)/(1.004996352)=23.8806837hours/day!
Over the past several billion years the length of the year has not deviated much from 365.25
days because of Kepler’s period for the orbit of a planet but, because the earth loses energy to
the moon, its days becomes longer over time by 0.0067 hours per million years which, we can
see from examining sedimentation band growth, which follows the lunar month. To get our
23.8806837 hours per day we have to go back about 18 million years:!
24-23.8806837=0.0067t!
t=17.80840299 million years!
This is the Miocene, a time which the Earth cooled slowly towards a series of ice ages,
between the Oligocene and the Pliocene defined by the boundaries of the cooler Oligocene
and the warmer Pliocene. See Messinian Salinity Crisis and Zarclean Flood. When the
dinosaurs roamed the Earth 70 million years ago the day was 23.7 hours long, about a half an
hour shorter. They seemed to have gone extinct about 65 million years ago when an asteroid
hit in the Yucatan, probably because we were moving through a place in the galaxy full of
debris, through which we pass every 27 million years. But asteroids in the asteroid belt pose a
threat as well, and that is why we catalog their trajectories. The asteroid belt is about 1AU thick
at 2.2 to 3.2 AU from the sun, the earth-sun separation being 1 AU. Carbon-12 the basis of life
as we know it is 12.01 g/mol. It is made in stars from Beryllium-8 is 8.0053051 g/mol. Thus we
have!
!
!
Delta x equal to 1 AU is both Earth and The Asteroids. Mars is at 1.52AU. Delta X cancels with
the Earth leaving Mars equal to carbon to beryllium, which is life, Does this say we need to be
able to colonize Mars to succeed as a species?!
1
6α
2
m
p
h4π r
2
p
Gc
= 1.004996352seconds
12
C
8
Be
=
Mars
Ear th
Δx
Δx = 1AU
of 8 31
!
Or, does it mean we need to put bases on the moon to mine Helium-3 as a clean, renewable
energy source.!
Not does this only mean the second is related to the earth orbital period, but now to other
things; By dividing the day into 24 hours, the hour into 60 minutes, and the minute into 60
seconds, the second is 1/86400 of day. By doing this we have a twelve-hour daytime at spring
and fall equinox on the equator, 12 being the most divisible number for its size (smallest
abundant number). That is to say that twelve is evenly divisible by 1,2,3,4,6 which precede it
and 1+2+3+4+6=16 is greater than twelve. As such there is about one moon per 30 days and
about 12 moons per year (per each orbit) giving us a twelve-month calendar. This is all further
convenient in that the moon and earth are in very close to circular orbits and the circle is evenly
divisible by 30, 45, 60, and 120 if we divide the circle into 360 degrees which are special angles
very useful to the workings of physics and geometry. Further, the 360 degrees of a circle are
about the 365 days of a year (period of one earth orbit) so as such the earth moves through
about a degree a day in its journey around the sun. Thus, through these observations down
through the ages since ancient times we have constructed the duration of a second wisely
enough to make a lot work together. Now we see 6 protons, which is carbon the core element
of biological life on the planet where all of this came together is deeply connected with the
second that defines it all. With this idea of proton-seconds describing hydrogen and carbon the
basis of life the hydrocarbons in a cycle of 6 with respect to one another, the motion of the
earth around the sun and moon around the earth, and the basis of geometry the 360 degree
circle, equation 1 connects them with the universal constant of gravitation, the speed of light,
the fine structure constant and Planck’s constant that characterizes the physics of the atom. I
really wonder if other star systems are connected so well their planets with their star as the
Earth with the Sun. We really can’t resolve planets around other stars because the stars are too
far and so bright compared to their orbiting planets and the planets have to be large enough to
be inferred by the motion they induce in the star they orbit. It is hard to do so with earth sized
planets that might harbor life. However that may change in the near future.!
Since the earth day gets longer at a rate of one hour per 0.0067 million years then, the earth
will be (24)(365.25)/(360)=24.35 hours per day when it goes through 360 days per year. This will
be in 0.35/0.0066=52.2388 million years or, the year 522,408,21.!
The constant described in the section Theory For Inertia, has had interesting applications with
asteroids, now I apply it again. Its Nature that most fascinates me is that the largest integer
value of six seconds predicts carbon and its smallest integer value of 1 second predicts
hydrogen. All other elements are predicted by values less than 1 and greater than zero. By
fractional values of seconds. So here I look at this considering the prebiotic path to amino
acids, the building blocks of life, which are NH3, CH4, CO2,…and I look at CH4 in that it is a
hydrocarbon the backbone of organic matter. Here are the calculations written out by hand
(next pages)…!
12
C =
4
He +
8
Be
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!
of 10 31
Thus we consider methane (CH4) since it is the simplest hydrocarbon that is one of the
prebiotic pathways to the amino acids, the building blocks of life:!
CH4 ==> C = 6 protons = 1 second!
H4 ==> H = 1 proton = 6 seconds!
H4 ==> 4 protons = 1.5 second!
We have: (6 protons)(4 protons) = 24 protons!
24 protons = 0.25 seconds!
Equation 3. !
= =!
=!
!
Which is chromium, element 24 which is 52 protons and neutrons or it is magnesium, which is
is element 12 and 24 protons and neutrons. This is the volume above a square and below the
surface f(x,y)=xy. See the figure below…!
!
4
9
⋅
1
α
2
⋅
1
m
p
h4π r
2
p
Gc
∫
6
0
∫
1
0
x yd x d y = 24protons
4
9
[6]
∫
6
0
∫
1
0
x yd x d y =
4
9
6
∫
6
0
∫
1
0
x yd x d y
4
9
6
∫
6
0
x
2
2
1
0
yd y
4
9
⋅ 6 ⋅
1
2
⋅ 18 = 24proton s
of 11 31
The Objective: To show that equation 1 is design.!
!
Where is hydrogen and 6 is the largest integer value of seconds predicting an
element, is carbon is the smallest integer value for seconds predicting an
element, and where these are the hydrocarbons which form the backbone of organic matter.!
To do this we need a working definition of design and I think here that Buckminster Fuller’s will
do fine from his work Synergetics: Design is that which gives you more degrees of freedom. He
coined his work as Design Science. In a sense we could say the Italian Renaissance artist and
inventor Leonardo Da Vinci was a design scientist. When he drew-up the first designs for a
helicopter, and a parachute, he was giving us more degrees of freedom.!
In the sense that this work strives to show Life as a Universal Natural Process we need a
definition for that as well. Here Buckminster Fuller works quite well again:!
“Humanity [is a] macro micro Universe-unfolding eventuation.”!
He further wrote:!
“In its complexities of design integrity, the Universe is technology…The technology evolved by
man is thus far amateurish compared to the elegance of non humanly contrived regeneration…
[It is] the gradual discovery of the function in Universe which humanity has been designed to
fulfill.”!
Working theorems from his work will help:!
610.01 By structure, we mean self-stabilizing pattern. The triangle is the only self-stabilizing
pattern.!
609.01 Any polygon with more than three sides is unstable. Only the triangle is inherently
stable.!
Essentially as the moon orbits the earth it makes 12 revolutions for each revolution of the Earth
around the sun which is 365.25 days. That is to say!
!
!
These are frequencies of!
!
!
In radians per day these are:!
1
t
1
α
2
m
p
h4π r
2
p
Gc
= 6protons
t
6
= 6seconds
t
1
= 1second
→
T
e
= 365.25d ay s
T
m
= 29.53059d ay s
f
e
= 0.002737851s
−1
f
m
= 0.033863191d a ys
−1
of 12 31
!
!
Thus the equations of their phases are:!
!
!
Where t is in days and 2.55605E-3=!
(radius lunar orbit)(radius earth orbit)=384,400km/(149,597,876km).!
We can say the frequency of the moon is 0.21/0.0172=12.21 times greater than that of the
earth. Thus we have the following plots of lunar phases to earth phases:!
Phases of the Moon And The Earth and Proton-Seconds
There are 12 moons in a year and 24 hours in a day. Divide twelve by 2 and we have 6, divide
24 by 2 and we have 12. We have:!
!
In that days=1, moons-12, hours=24.!
At this point we bring-up that remarkable fact the the moon perfectly eclipses the sun. This is
because:!
ω
e
= 0.0172
ω
m
= 0.21
y
e
= cos(ω
e
t)
y
m
= (2.57E − 3)cos(ω
m
t)
moons
2
hours ⋅ da ys
= 6
of 13 31
!
!
Which are approximately equal. As well we can look at it as:!
!
!
Which are about the same as well. The interesting thing is that since our ratios are around
0.0025 and 0.0045, then…!
!
I say this is interesting because this the ratio of the precious metal gold (Au) to that of silver
(Ag) by molar mass:!
!
We have:!
!
!
Taking the average between these:!
!
Where, is the lunar orbit, is the solar radius, is the earth orbit, and is the radius of
the moon. What this means is that the moon perfectly eclipses the sun because the solar
radius is 1.8 times greater than the lunar orbital radius. And interestingly gold is yellow, silver is
silver and the sun is gold, and the moon is silver, the moon perfectly eclipses the sun allowing
us to study its outer atmosphere, and the exact number of moons in a year is:!
(lun ar − orbit)
(ear th − orbit)
=
384,400k m
149,592,870k m
= 0.00257
(lun ar − ra dius)
solar − ra dius
=
1,738.1
696,00
= 0.0025
(lun ar − ra dius)
(lun ar − orbit)
=
(1,738.1)
(384,400)
= 0.00452
solar − ra dius
ear th − orbit
=
696,000
149,597,870
= 0.00465
0.0045
0.0025
=
9
5
= 1.8
Au
Ag
=
196.97
107.87
= 1.8
(lun ar − ra dius)(ear th − orbit)
(lun ar − orbit)
2
= 1.759577590
(solar − ra dius)
2
(ear th − orbit)(lun ar − ra dius
= 1.863
1
2
⋅
(
r
2
m
⋅ R
2
⊙
+ r
2
e
R
2
m
r
2
m
⋅ r
e
⋅ R
m
)
=
Au
Ag
r
m
R
⊙
r
e
R
m
of 14 31
!
x=12.3685304!
Is the consecutive integers 0, 1, 2, 3, 4. 5, 6,…8 only missing the 7. Where the equation!
square moons per 24-hour day,…!
Is concerned, I believe it is deeply interacting throughout nature with the equation for proton-
seconds:!
!
Because !
And there are 60 minutes in an hour, and 60 seconds in a minute. We can equate them
because while:!
!
Has units of mass, dividing it by gives a number of protons, but but as well you can think of
the cancelling with the mass leaving pure number, and in!
!
You can think of 12 moons being the number of new moons that appear in the sky per time of
the journey of the sun North and back South again marking the seasons. We can divide each
day into 24 units and so it may be that there is some dynamic behind:!
!
Meaning:!
!
365.25
x
= 29.53059d ay s
(12m oon s)
2
24hours ⋅ 1da y
= 6
1
t
1
α
2
m
p
h4π r
2
p
Gc
= 6protons
t
1
= 1second
1
t
1
1
α
2
⋅
h4π r
2
p
Gc
m
p
m
p
(12m oon s)
2
24hours ⋅ 1da y
= 6
1
t
1
α
2
m
p
h4π r
2
p
Gc
=
(12m oon s)
2
24hours ⋅ 1da y
6proton s =
(12m oon s)
2
24hours ⋅ 1da y
= C
of 15 31
Carbon the basis of life as we know it. Indeed we can make sense of 24 hours in a day. We
could divide the the 365 day year that we observe in the sun’s journey across the sky by 24
and we get:!
365/24=15.208!
Not a whole number, but we could try!
360/24=15!
Even. Then as such we divide the circle into 360 such units, and we have our 60 minutes and
60 seconds and our six protons:!
360/60=6!
The 360 is further evenly divisible by 30, 45, 60, 90, 120, and 180. The angles 30, 45, 60, and
90 are angles in what mathematicians call special triangles because a 45-45-90 is!
!
A 30-60-90 is!
!
!
And further:!
!
Is the golden ratio. Where 36=360/10 and ten is our decimal system of counting (base 10
perhaps because we have 10 finger to count on). I like to write this as!
Equation 4. !
Where n=3, 4, 5, ,6 ,…!
With 360 degrees the earth moves through about one degree a day in its journey around the
sun.
cos45
∘
= 2 /2
cos30
∘
= 3 /2
cos60
∘
= 1/2
cos36
∘
= Phi =
5 + 1
2
2 cos(π /n) = 1, 2, Φ, 3, . . .
of 16 31
Conclusion 1
While the equations!
Produce the elements for values less than 1 and greater than zero (fractional values) 1 second
produces hydrogen and 6 seconds produces carbon, and these are the lower and upper limits
for integers producing a whole number of elements, putting hydrocarbons, the backbone of life
chemistry as having components out of phase with one another by cycles of six in seconds.
Seconds come from having divided the calendar according to the cycles of the moon, the spin
frequency of the Earth, and its orbital period such that they all mesh with one another
approximately according to special angles recurrent throughout mathematics and physics.
These two equations have G, the universal gravitational constant among other parameters
associated with the physics of everything. The cycle of six for the phase shift from hydrogen to
carbon may be characteristic of physical forces because six-fold symmetry is often more
characteristic of the physical where five-fold is more characteristic of life as was pointed-out by
the Russian Scientist Shubnikov. One might look at this as something physical, like snowflakes
form six points (six-fold radial symmetry) due to acceleration of an ice particulate against air
gathering moisture and freezing it as it falls under earth gravity, and five-fold symmetry, as in
the large variety of five-petaled flowers (five-fold radial symmetry) because it has the golden
ratio in it, the ratio of the chord of a regular pentagon to its side, and six-fold symmetry
embodies unity as the radius of a regular hexagon is equal to its side and the golden ratio is
often used by life for closest packing, as occurs in the hundreds of petals in a rose spiraling in
as a golden spiral to its center. Ultimately there is a mystery associated with the phases of the
moon and earth with their highly organizable structures associated with the basis of
mathematics and physics.!
1
α
2
m
p
h4π r
2
p
Gc
= (6protons)(1second )
1
α
2
m
p
h4π r
2
p
Gc
= (1proton)(6secon d s)
of 17 31
The Geometrical Explanation of Seconds
Here we will talk about the equation 1 (Which refers to the next eight pages of illustrations):!
In so far as the second unifies carbon (6 protons) with hydrogen (1 proton) through the unit of a
second as the hydrocarbons the backbone of life. We have suggested the second is important
as well in terms of the phases of the moon and the earth and that these determined the
calendar system we use. We further suggested there is a connection of this to the structure
found in geometry, and this is what we want to explore further, here. We ended with all of this
can be compactly written as:!
!
Where n=3, 4, 5, ,6 ,…!
We could evaluate this for n equal to other integers, or even the numbers, but these produce
the special triangles, and geometries we are most interested. Thus we will begin by pointing
out that!
3 X 4 X 5 X 6 = 360!
The amount of degrees into which we divide a circle and that, as such it approximates the the
number of days in a year (1 revolution of the earth around the sun) and thus we see that
(equation 4)!
Equation 4. !
Represents days as well (The earth moves through about 1 degree a day in its orbit around the
sun) by solving it for n:!
Equation 5. !
Where,!
!
Which correspond respectively to:!
!
Which are the unit triangle, the square, the regular pentagon, and the regular hexagon:!
1
t
1
α
2
m
p
h4π r
2
p
Gc
= 6protons
2 cos(π /n) = 1, 2, Φ, 3, . . .
2 cos(π /n) = 1, 2, Φ, 3, . . .
days = cos
−1
(y/2)
y = 1, 2,
5 + 1
2
, 3, . . .
n = 3,4,5,6,...
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#
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#
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!
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!
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!
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!
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!
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!
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The Derivative
We take our function!
And write it:!
!
Then take the derivative:!
!
This is a right triangle with hypotenuse 2, and height x and with base :!
To get , then . To get , then , and to get then .
To get then . Thus our function is equation four written:!
!
Where:!
!
n=3,4,5,6,,…!
=!
is the rate of change of a triangle (x=1). 0.85 is the rate of change of a regular pentagon
( ). is the rate of change of a square ( ). 1 is the rate of change of a regular
hexagon ( ). #
days = cos
−1
(y/2)
y = cos
−1
(x /2)
dy
dx
= −
1
4 − x
2
4 − x
2
θ = 60
∘
x = 3
θ = 45
∘
x = 2
θ = 30
∘
x = 1
θ = 36
∘
4 − x
2
2
= Φ
y = 2cos(θ (n))
θ (n) =
π
n
=
180
∘
n
dy
dx
= −
1
4 − x
2
1/ 3
x = Φ
2
2
x = 2
x = 3
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!
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!
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!
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Conclusion 2
We see that!
!
!
Is x as a function of n, and that n=3 is a unit triangle, n=4 a unit square, n=5 is a unit pentagon,
and n=6 is a unit hexagon. Thus if n=3 we have the unit triangle is Earth, the unit square is
Mars, the unit pentagon is n=5 which not only is this shape not a member of the regular
tessellators, n=5 is the asteroid belt, which is a location in the solar system where a planet
cannot form. We then proceed to Jupiter, which is n=6, the most massive planet in the solar
system which carries the majority of its mass. dy/dx is the the change in days with respect to
planetary number.#
y = cos
−1
(x /2)
dy
dx
= −
1
4 − x
2
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The Author
